# Questions tagged [covariance]

Questions about covariance, a measure of (linear) association between two random variables.

1,096 questions
5 views

### what is the covariance matrix of 4x2 data points?

I do not know how to calculate the covariance matrix for a 4x2 data points The points are as followed: X | Y -2 | -1 -1 | 0 1 | 0 2 | 1 Can I get some ...
10 views

### Rank-1 modification of correlation matrix

I inherited a problem at work and have trouble wrapping my head around it. It doesn't help I'm insufficiently sophisticated with linalg. Help with or pointing to appropriate literature will be much ...
59 views

23 views

### Given that $X_{i+1} = \rho X_{i}$, determine the dispersion matrix $Var[\textbf{X}]$

If $X_{1},X_{2},\ldots,X_{n}$ are random variables and $X_{i+1} = \rho X_{i}$ $(i = 1,2,\ldots,n)$, where $\rho$ is constant, and $\mathrm{Var}[X_1] = \sigma^2$, find $\mathrm{Var}[X]$. MY ATTEMPT ...
17 views

### Distribution/Variance of correlated squared normal random variables

If $X_{1}, X_{2}, \ldots, X_{N}$ are identically distributed normal random variables with mean $0$ and variance $\frac{(N+3)D\sigma^{2}}{N}$, then I want to calculate the distribution, or at least the ...
216 views

19 views

### Show diagonal covariance does not guarantee independence.

I was trying to show that if two variables have diagonal covariance, this does not necessarily guarantee their independence. For this, I was using an example where $x \sim U(-1,1)$ and $y=X^{2}$ to ...
19 views

### error covariance of MMSE estimator relation to other error covariance estimators

I'm trying to prove the following: let $\Lambda_{e}$ be the error covariance of an estimator $\,\hat{\theta}(y)$ of $\,\theta$ based on $\,y$. I want to show that the error covariance of MMSE ...
72 views

### Log det of covariance and entropy

I understand log of determinant of covariance matrix bounds entropy for gaussian distributed data. Is this the case for non gaussian data as well and if so, why? What does Determinant of Covariance ...
152 views

### Transformation of a valid covariance matrix

Suppose we have a normally distributed random variable $\boldsymbol{X}$ with covariance matrix $\Sigma$, which is symmetric positive definite. Now if I multiply $\boldsymbol{X}$ by some matrix A, the ...
9 views

### Determinants of the covariance between two random variables

I want to know if the covariance between two random variables is always determined by the two terms of its formula or if it can be only determined by the E(XY). If E(Y)=0, then Cov(X,Y)=E(XY)-E(X)E(Y)...
26 views

### Given two random vectors, determine the dispersion matrix $Var[\textbf{X}]$.

Let $\textbf{X} = (X_{1},X_{2},\ldots,X_{n})^{\prime}$ be a vector of random variables, and let $Y_{1} = X_{1}$ and $Y_{i} = X_{i}-X_{i-1}$ where $i = 2,3,\ldots,n$. If $Y_{i}$ are mutually ...
29 views

### Calculate covariance given correlation, problem with percentages

The question is: find the covariance of ABC stock returns with the original portfolio returns. Pretty straightforward. However I get confused working between percentages and units. The ...
707 views

### Covariance of Martingales

I have proven that Martingales have orthogonal increments. From this I need to show that $\operatorname{Cov}[M(t),M(s)]$ relies only on $\min\{s,t\}$. I used the expected value definition of ...
42 views

### Covariance matrix of uniform distribution in $L^p$ Euclidean ball [closed]

Let $$X=\operatorname{Unif}\left(\left\{x:\ \sum_{i=1}^n |x_i|^p\le 1\right\}\right)$$ Is there some method to calculate the covariance matrix of $X$? Thank you!
14 views

### truncation degree of decomposed covariance matrix

I have a covariance matrix of a standardized data set. Doing a singular value decomposition i find near zero singular values and would therefore like to truncate it. I know of Picard plots which ...
34 views

### Using Cholesky decomposition to compute covariance matrix determinant

I am reading through this paper to try and code the model myself. The specifics of the paper don't matter, however in the authors matlab code I noticed they use a Cholesky decomposition instead of ...
113 views

### Constructing a probability measure on the Hypercube with given moments

Let $H = [-1, 1]^d$ be the $d$-dimensional hypercube, and let $\mu \in \text{int} H$. Under these conditions, I can explicitly construct a tractable probability measure $P$, supported on on $H$, ...
50 views

26 views

### Covariance of square root for two bins of a multinomial

Take $(X_1, \dots, X_k) \sim Multinomial(n, (p_1, \dots, p_k))$. Do we have a closed form expression for $\mathbb{E}[\sqrt{X_i X_j}], i\neq j$ ?
766 views

### Variance of $3$-dimensional vectors

I am currently optimizing some code and thus, I want to replace an inefficient OpenCV function, which calculates a covariance matrix. The thing is, that I only need the trace of this covariance matrix,...
17 views

### Normal Random Variables that all correlate with a single time series but not necessarily with each other

I have a sequence of normally distributed random variables. Let's call it $S_1$. I want to generate 4 more series, each of which has its own correlation with $S_1$ and its own variance. Let's call ...
27 views

### Finding $\text{Cov}(X,Y)$ when $(X,Y)$ has joint density $\frac{1}{2}\sin(x+y)\mathbf1_{0\le x,y\le\pi/2}$

Joint probability density: P_{x,y}(x,y) \begin{cases} \frac{1}{2}\sin(x + y) & , \text{if}\ 0\leq x \leq \frac{\pi}{2}, 0 \leq y \leq \frac{\pi}{2} \\ 0 & ...
12 views

### Scalar product induced by covariance matrix

Suppose that $n$-dimensional random vector $Y$ has covariance matrix $\Sigma$. It is well known that for any $a\in\mathbb{R}^n$ we have \begin{align} var(a^TY)=a^T\Sigma a. \end{align} Is there any ...
21 views

### Variance and covariances from linear mixed model for power simulation using R

I am working with longitudinal data where the outcome is the number of steps per minute. My LMM fit would look like: ...
117 views

### Explicitly proving that the Hamiltonian is Lorentz covariant

I want to show explicitly that the Hamiltonian $$H = -\Omega V + \int d\tilde{\textbf{k}}\ \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) )$$ is Lorentz ...